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Formally, Bayesian networks are Directed Acyclic Graph s whose nodes represent variables, and whose arcs encode conditional independencies between the variables. Nodes can represent any kind of variable, be it a measured parameter, a Latent Variable or a hypothesis. They are not restricted to representing Random Variable s, which represents another " Bayesian " aspect of a Bayesian network. Efficient algorithms exist that perform Inference and learning in Bayesian networks. Bayesian networks that model sequences of variables (such as for example Speech Signals or Protein Sequences ) are called Dynamic Bayesian Networks . Generalizations of Bayesian networks that can represent and solve decision problems under uncertainty are called Influence Diagrams . DEFINITIONS AND CONCEPTS If there is an Arc from node ''A'' to another node ''B'', ''A'' is called a ''parent'' of ''B'', and ''B'' is a ''child'' of ''A''. The set of parent nodes of a node ''X''i is denoted by parents(''X''i). A directed acyclic graph is a Bayesian Network relative to a set of variables if the joint distribution of the node values can be written as the product of the local distributions of each node and its parents: : If node ''X''''i'' has no parents, its local probability distribution is said to be ''unconditional'', otherwise it is ''conditional''. If the value of a node is ''observed'', then the node is said to be an ''evidence'' node. Reading independencies and ''d''-separation The graph encodes independencies between variables. ''. D-separation is defined as follows. A path p is said to be d-separated (or blocked) by a set of nodes Z if and only if 1. p contains a chain p -> m -> j or a fork i <- m -> j such that the middle node m is in Z, or 2. p contains an inverted fork (or collider) i -> m <- j such that the middle node m is not in Z and no descendant of m is in Z. A set Z is said to d-separate x from y in a Directed Acyclic Graph G if all paths from x to y in G are d-separated by Z. The 'd' in d-separation stands for 'directional', since the behavior of a three node link on a path depends on the direction of the arrows in the link. Causal Bayesian networks A Bayesian network is a carrier of the conditional independencies of a set of variables, not of their causal connections. However, causal relations can be modelled by the closely related Causal Bayesian Network . The additional semantics of the causal Bayesian networks specify that if a node ''X'' is actively caused to be in a given state ''x'' (an operation written as ''do(x)''), then the probability density function changes to the one of the network obtained by cutting the links from ''X'''s parents to ''X'', and setting ''X'' to the caused value ''x'' (Pearl, 2000). Using this semantics, one can predict the impact of external interventions from data obtained prior to intervention. EXAMPLE Suppose that there are two reasons which could cause grass to be wet: either the sprinkler is on or it's raining. Also, suppose that the rain has a direct effect on the use of the sprinkler (namely that when it rains, the sprinkler is usually not turned on.) Then the situation can be modelled with the adjacent Bayesian network. All three variables have two possible values T (for true) and F (for false). The joint probability function is: |
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